CN106294283A - Temporal basis functions fast algorithm based on Taylor series expansion - Google Patents

Abstract

The invention discloses a kind of temporal basis functions fast algorithm based on Taylor series expansion.Traditional temporal basis functions method cannot solve large-scale electromagnetic problem due to the restriction of calculating time and memory consumption.And developing fast algorithm is the only way solving Practical Project problem.The present invention be utilize the Taylor series expansion of Green's function in free space reconstruct polymerization, shift, the form that configures and then realize the speed-up computation that matrix vector is taken advantage of, the present invention needs less calculating internal memory and calculating time for solving metal complex target scattering problems.And program relatively easy being easily achieved, there is the strongest practical engineering application and be worth.

Description

Temporal basis functions fast algorithm based on Taylor series expansion

Technical field

The present invention relates to metal target Transient Em Fields Scattering numerical calculation of fan properties technology, belong to the quick computing technique field of electromagnetic characteristic of scattering, a kind of temporal basis functions fast algorithm based on Taylor series expansion.

Background technology

Along with going deep into of Computational electromagnetics area research, traditional frequency domain method can not meet needs.With the opportunity that develops into of computer hardware technology, people are progressively provided with directly in the computational analysis ability of the time domain transient electromagnetic field to having broadband character, it is achieved thereby that to physical quantity and physical phenomenon more deeply, understand more intuitively.Compared with frequency domain method, the electromagnetic property solving target in time domain is possible not only to disclose target and electricity intuitively

The mechanism that magnetic wave interacts, and logical too small amount of calculating is obtained with the wideband information of target, and this has obvious advantage in wideband electromagnetic problem, Transient Electromagnetic case study.

Compared with time domain approach (FDTD, FETD etc.) based on the differential equation, time domain approach based on integral equation has obvious advantage in the unknown number quantity solved, this is because integral equation utilizes Green's function to set up the relation of source and field, domain is on border, and the quantity of discrete rear unknown number is directly proportional to boundary areas.Secondly, integral Equation Methods meets radiation boundary condition automatically, it is not necessary to force absorbing boundary, and absorbing boundary is necessary to method based on the differential equation.

Due to computer level and the restriction of hardware, traditional temporal basis functions method cannot solve large-scale electromagnetic problem.And developing fast algorithm is the only way solving Practical Project problem.The research of temporal basis functions fast algorithm was carried out at nearly 20 years, and the most foremost two kinds: one is Time Domain Planar ripple algorithm (PWTD), and two is time-domain adaptive integral equation (TD-AIM).Three dimensional object model, the amount of calculation of TD-AIM and amount of storage are respectivelyWithThe inefficiency of its process objective visible.Its principle of PWTD algorithm is similar to FMM algorithm, but is because the existence of time variable so that PWTD algorithm is more complicated, it is achieved get up more difficult.And the bad adaptability of this algorithm, when in order to solve the challenge such as medium or lossy medium, in addition it is also necessary to algorithm is carried out special modification.

Summary of the invention

It is an object of the invention to provide the fast method of a kind of efficient, stable time domain electromagnetic characteristic analyzing metal target.Due to far-field portion is used Taylor series expansion reconstruct polymerization, shift, the form that configures and then realize the speed-up computation that matrix vector is taken advantage of, the present invention is for solving the metal complex target scattering problems less calculating internal memory of needs and calculating time.And program relatively easy being easily achieved.

The technical solution realizing the object of the invention is: a kind of temporal basis functions fast algorithm based on Taylor series expansion, and step is as follows:

The first step, sets up time-domain electromagnetic integral equations.Metal target is under the effect of incident electromagnetic wave, the surface of metallic cylinders can produce face sensitive surface electric current, faradic it is continually changing, can outwards give off electromagnetic field, the electromagnetic field produced by faradic current is exactly scattered field, and the boundary condition utilizing incident electromagnetic field and scattering field to meet in metal surface sets up time-domain electromagnetic integral equations.

Second step, by the sub-scattering object packet obtained discrete on scattering object surface.Mutual coupling between the sub-scattering object of any two or self coupling be divided into according to the position relationship of they place groups near field group to and far field group pair.When they are near field groups pair, use direct numerical computations.And when they are far field group pair, then use Taylor series expansion to become polymerization-transfer-collocation method to calculate.

3rd step, near field impedance matrix calculus, by metal surface electric current density space basic function and time base function expansion, and in spatial domain, carry out the gold test of gal the Liao Dynasty, time domain carries out Point matching and obtains matrix element value.

4th step, far-field portion uses Taylor series expansion to reconstruct polymerization, the form shifting, configuring.

5th step, matrix equation solves and the calculating of electromagnetic scattering parameter.The mode utilizing time recursion solves the current coefficient in each moment.The scattered field contribution sources in somewhere, space of each moment is divided into contribution and the contribution in group source, far field in group source, near field.The former is directly multiplied with current coefficient by near field matrix and obtains, and the latter becomes polymerization-transfer-collocation method quickly to calculate by Taylor series expansion.Solution by iterative method is used to go out final scattering current coefficient.And solve Radar Cross Section according to reciprocal theorem.

Compared with prior art, its remarkable advantage is the present invention: (1) can quickly analyze complicated TV university target time domain electromagnetic characteristic；(2) far field calculates based on Taylor series expansion, it is to avoid complicated time-frequency domain conversion that PWTD algorithm needs, k-space integrals etc. operate, and program relatively easy being easily achieved；(3) suitability is high, only can need to make less change and just can be applicable to solve medium, in the analysis that has the challenge such as consumption, dispersion.

Accompanying drawing explanation

Fig. 1 is the multilamellar being applied in temporal basis functions fast algorithm based on the Taylor series expansion packet schematic diagram of the present invention.

Fig. 2 is the nearly far field relation schematic diagram being applied in temporal basis functions fast algorithm based on Taylor series expansion of the present invention.

Fig. 3 is the field source basic function position vector distribution schematic diagram being applied in temporal basis functions fast algorithm based on Taylor series expansion of the present invention.

Fig. 4 is dual station RCS comparison diagram at the simple guided missile model different frequent points being applied in temporal basis functions fast algorithm based on Taylor series expansion of the present invention.(a)30MHz；(b)150MHz；(c)270MHz.

Detailed description of the invention

Below in conjunction with the accompanying drawings the present invention is described in further detail.

A kind of temporal basis functions fast algorithm based on Taylor series expansion of the present invention, step is as follows:

The first step, sets up time-domain electromagnetic integral equations.

Metal target is at incident electromagnetic wave { E^inc(r,t),H^inc(r, t) } effect under, (r, t), faradic is continually changing, can outwards give off electromagnetic field, faradic current the electromagnetic field produced is exactly scattered field { E to produce face sensitive surface electric current J on the surface of metallic cylinders^sca(r,t),H^sca(r, t) }, the boundary condition utilizing incident electromagnetic field and scattering field to meet in metal surface sets up time-domain electromagnetic integral equations.

$\begin{array}{c}\mathrm{\α}\hat{n}\left(r\right)\×\hat{n}\left(r\right)\×E^{inc}(r,t)+(1-\mathrm{\α})\mathrm{\η}\hat{n}\left(r\right)\×H^{inc}(r,t)\\ ={\mathrm{\αL}}_e\left\{J(r,t)\right\}+\mathrm{\η}(1-\mathrm{\α})L_h\left\{J(r,t)\right\}\end{array}---\left(1\right)$

$\begin{array}{c}{L}_e\left\{J(r,t)\right\}=\hat{n}\left(r\right)\×\hat{n}\left(r\right)\×\frac{\mathrm{\μ}}{4\mathrm{\π}}\∫{\∫}_{s}dS\frac{1}{R}\frac{\∂J(r^{\′},\mathrm{\τ})}{\∂t}-\\ n\left(r\right)\×\hat{n}\left(r\right)\×\frac{1}{4\mathrm{\π}\mathrm{\ϵ}}\∫{\∫}_{s}dS{\∫}_0^{\mathrm{\τ}}\frac{{\▿}^{\′}\·J(r^{\′},\mathrm{\τ})}{R}dt\end{array}---\left(2\right)$

$L_h\left\{J(r,t)\right\}=\frac{1}{2}J(r,t)-\hat{n}\left(r\right)\×\frac{1}{4\mathrm{\π}}\∫{\∫}_{s}dS\▿\×\frac{J(r^{\′},\mathrm{\τ})}{R}---\left(3\right)$

WhereinBeing the outer normal vector of unit, α value is the real number between 0～1, μ and ε is pcrmeability and the dielectric parameter of free space respectively.η is the natural impedance of free space.R=| r-r'|, c are the lighies velocity in free space, and τ=t-R/c is time delay.

Second step, by the sub-scattering object packet obtained discrete on scattering object surface.

We use for reference the thought inside MLFMM being grouped space basic function, and the interaction of basic function is converted to the interaction of group.Assume that space has a cube that just whole target object can be surrounded, this cube is divided into 8 sub-cubes, then each sub-cube is being divided into 8 less cubes, as shown in Fig. 1 (a-d).The like, until reaching the threshold value pre-set, stop dividing.Definition ground floor is to divide the thinnest one layer, if comprising basic function in the cube in ground floor, is defined as ground floor group, for other layer of group, the like.

Mutual coupling between the sub-scattering object of any two or self coupling be divided into according to the position relationship of they place groups near field group to and far field group pair.For each layer, we can set a parameter beta, 4 ＜ β ＜ 6.First from top N_lEach group set about divide, if the distance between two group switching centres more than regulation threshold value, just it is referred to as far field group pair；Then, in its sublayer, group is more than the threshold value of this layer of regulation to center distance, and the two group is not the most divided into N simultaneously_lThe far field group pair of layer, then just specify such group to the far field group pair for this layer；In accordance with the above the like, for each layer, we can be obtained by the information of its far field group pair, as shown in Figure 2.When they are near field groups pair, use direct numerical computations.And when they are far field group pair, then use Taylor series expansion to become polymerization-transfer-collocation method to calculate.

3rd step, near field impedance matrix calculus.

Metal surface circuit current density J, (r, t) with RWG basic function Λ_nR () is as space basic function, Based on Triangle Basis T_jT () makees time base function expansion:

$J(r,t)=\underset{n=1}{\overset{N_S}{\Σ}}\underset{j=1}{\overset{N_t}{\Σ}}{I}_{n,j}{\mathrm{\Λ}}_n\left(r\right)T_j\left(t\right)---\left(4\right)$

Wherein N_sIt is the number of the RWG basic function that scattering object comprises, N_tIt is the number of time basic function, I_n,jIt it is the coefficient of time basic function on the n-th RWG basic function jth time step.Formula (4) is substituted into (1), spatially carries out Galerkin test, carry out Point matching in time, finally give following matrix equation:

$Z^0{I}^i=V^i-\underset{j=1}{\overset{i-1}{\Σ}}{Z}^{i-j}{I}^j---\left(5\right)$

$Z^{i-j}={\mathrm{\αZ}}_E^{i-j}+(1-\mathrm{\α})Z_M^{i-j}---\left(6\right)$

${\[Z_E^{i-j}\]}_{mn}=\left\{\begin{array}{c}\frac{{\mathrm{\μ}}_0}{4\mathrm{\π}}{\∫}_{S_m}{\∫}_{S_n}{\mathrm{\Λ}}_m\left(r\right)\·{\mathrm{\Λ}}_n\left(r^{\′}\right)\frac{{\∂}_{\mathrm{\τ}}{T}_j(i\mathrm{\Δ}t-R/c)}{R}{\mathrm{ds}}^{\′}ds+\\ \frac{1}{4{\mathrm{\π\ϵ}}_0}{\∫}_{S_m}{\∫}_{S_n}\▿\·{\mathrm{\Λ}}_m\left(r\right)\▿\·{\mathrm{\Λ}}_n\left(r^{\′}\right)\frac{{\∂}_{\mathrm{\τ}}^{-1}{T}_j(i\mathrm{\Δ}t-R/c)}{R}{\mathrm{ds}}^{\′}ds\end{array}\right.---\left(7\right)$

${\[Z_M^{i-j}\]}_{mn}=\left\{\begin{array}{c}\frac{1}{2}{\∫}_{S_m}{\mathrm{\Λ}}_m\left(r\right)\·{\mathrm{\Λ}}_n\left(r^{\′}\right)T_j\left(i\mathrm{\Δ}t\right)ds-\frac{1}{4\mathrm{\π}}{\∫}_{S_m}{\mathrm{\Λ}}_m\left(r\right)\\ \{\hat{n}\left(r\right)\×{\∫}_{S_n}\left\{\begin{array}{c}{\∂}_{\mathrm{\τ}}{T}_j(i\mathrm{\Δ}t-R/c){\mathrm{\Λ}}_m\left(r^{\′}\right)\×\frac{R}{R^2}+\\ T_j(i\mathrm{\Δ}t-R/c){\mathrm{\Λ}}_m\left(r^{\′}\right)\×\frac{R}{R^3}\end{array}\right\}\}{\mathrm{ds}}^{\′}ds\end{array}\right.---\left(8\right)$

4th step, far-field portion uses Taylor series expansion to reconstruct polymerization, the form shifting, configuring.

Now by the source signal J calculated at source point r'_n(r' t) describes the ultimate principle of this algorithm to the radiation contributions at site r.The space basic function assuming source point r' place is Λ_n(r'), therefore source signal J at r'_n(r', t) can be unfolded as follows:

$J_n(r^{\′},t)=\underset{j=1}{\overset{N_t}{\Σ}}{I}_{n,j}{\mathrm{\Λ}}_n\left(r^{\′}\right)T_j\left(t\right)={\mathrm{\Λ}}_n\left(r^{\′}\right)g_n\left(t\right)---\left(9\right)$

In practical operation, we need to be decomposed into source signal persistent period shorter segmentation subsignal, it is ensured that the persistent period of every cross-talk signal meets constraints.Source signal J_n(r' t) can be broken down into L section continuous print subsignal J_n,l(r', t), the persistent period of each cross-talk signal is T_s=(M_t+1)Δt.Source signal can be to be written as form:

$J_n\left(r^{\′},t\right)=\underset{l=1}{\overset{L}{\Σ}}{J}_{n,l}(r^{\′},t)=\underset{l=1}{\overset{L}{\Σ}}{\mathrm{\Λ}}_n\left(r^{\′}\right)g_{n,l}\left(t\right)---\left(10\right)$

The checkout area that at so source point r', l cross-talk signal produces at site r is:

$\begin{array}{c}\⟨{\mathrm{\Λ}}_m\left(r\right),n_m\×H_{n,l}(r,t)\⟩\\ =\frac{1}{4\mathrm{\π}}{\∫}_{S_m}{\mathrm{\Λ}}_m\left(r\right)\·n_m\×\left\{{\∫}_{S_n}{\mathrm{dS}}^{\′}\{\[{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right){\mathrm{\Λ}}_n(r^{\′})\]\×\frac{R}{{\mathrm{cR}}^2}+\[g_{n,l}\left(\mathrm{\τ}\right){\mathrm{\Λ}}_n(r^{\′})\]\×\frac{R}{R^3}\}\right\}dS\\ \≈{\frac{1}{4\mathrm{\π}}}_m\left({\∫}_{S_m}{\mathrm{\Λ}}_m\left(r\right)dS\·n_m\×{\∫}_{S_n}{\mathrm{\Λ}}_n\left(r^{\′}\right)d{S}^{\′\×R}\right)\[\frac{1}{{\mathrm{cR}}^2}{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right)+\frac{1}{R^3}{g}_{n,l}\left(\mathrm{\τ}\right)\]\\ =\frac{1}{4\mathrm{\π}}(m_m\·n_m\×m_n\×R)\[\frac{1}{{\mathrm{cR}}^2}{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right)+\frac{1}{R^3}{g}_{n,l}\left(\mathrm{\τ}\right)\]\end{array}---\left(11\right)$

$\begin{array}{c}\⟨{\mathrm{\Λ}}_m\left(r\right),E_{n,l}(r,t)\⟩\\ ={\∫}_{S_m}{\∫}_{S_n}{\mathrm{\Λ}}_m\left(r\right)\·{\mathrm{\Λ}}_n\left(r^{\′}\right)\frac{{\mathrm{\μ}}_0{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right)}{4\mathrm{\π}R}{\mathrm{ds}}^{\′}ds+{\∫}_{S_m}{\∫}_{S_n}\▿\·{\mathrm{\Λ}}_m\left(r\right)\▿\·{\mathrm{\Λ}}_n\left(r^{\′}\right)\frac{{\∂}_{\mathrm{\τ}}^{-1}{g}_{n,l}\left(\mathrm{\τ}\right)}{4{\mathrm{\π\ϵ}}_{0}R}{\mathrm{ds}}^{\′}ds\\ \≈{\∫}_{S_m}{\mathrm{\Λ}}_m\left(r\right)ds\·{\∫}_{S_n}{\mathrm{\Λ}}_n\left(r^{\′}\right){\mathrm{ds}}^{\′}\frac{{\mathrm{\μ}}_0{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right)}{4\mathrm{\π}R}+{\∫}_{S_m}\▿\·{\mathrm{\Λ}}_m\left(r\right)ds{\∫}_{S_n}\▿\·{\mathrm{\Λ}}_n\left(r^{\′}\right){\mathrm{ds}}^{\′}\frac{{\∂}_{\mathrm{\τ}}^{-1}{g}_{n,l}\left(\mathrm{\τ}\right)}{4{\mathrm{\π\ϵ}}_{0}R}\\ =\frac{{\mathrm{\μ}}_0}{4\mathrm{\π}}{m}_m\·m_n\frac{{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right)}{R}\±\frac{l_m{l}_n}{4{\mathrm{\π\ϵ}}_0}\frac{{\∂}_{\mathrm{\τ}}^{-1}{g}_{n,l}\left(\mathrm{\τ}\right)}{R}\end{array}---\left(12\right)$

Wherein $m_m={\∫}_{S_m}{\mathrm{\Λ}}_m\left(r\right)ds,m_n={\∫}_{S_n}{\mathrm{\Λ}}_n\left(r^{\′}\right){\mathrm{ds}}^{\′}.$

It is now assumed that source point r_nWith site r_mLaying respectively in two groups, as it is shown on figure 3, two groups are called source group and field group, group switching centre is respectively r_iAnd r_j, the vector between field source basic function can be expressed as:

R=r_mi+r_ij-r_nj=R_m-R_n (13)

Here, r_ij=r_i-r_j, r_mi=r_m-r_i, r_nj=r_n-r_j R_m=r_mi+r_ij/ 2, R_n=r_nj-r_ij/2

Utilize Taylor series expansion can obtain following expression:

$\begin{array}{c}{R}^{\mathrm{\α}}={(R\·R)}^{\frac{\mathrm{\α}}{2}}={\[(r_{mi}+r_{ij}-r_{nj})\·(r_{mi}+r_{ij}-r_{nj})\]}^{\frac{\mathrm{\α}}{2}}\\ =r_{ij}^{\mathrm{\α}}{\[1+(\frac{2r_{mi}\·{\hat{r}}_{ij}}{r_{ij}}+\frac{r_{mi}^2}{r_{ij}^2})+(\frac{2r_{nj}\·{\hat{r}}_{ji}}{r_{ij}}+\frac{r_{nj}^2}{r_{ij}^2})-\frac{2r_{mi}\·{\hat{r}}_{nj}}{r_{ij}^2}\]}^{\frac{\mathrm{\α}}{2}}\\ \≈R_m^{\left(\mathrm{\α}\right)}+R_n^{\left(\mathrm{\α}\right)}\end{array}---\left(14\right)$

$R_m^{\left(\mathrm{\α}\right)}=r_{ij}^{\mathrm{\α}}\[\frac{1}{2}+\mathrm{\α}\left(\frac{r_{mi}\·{\hat{r}}_{ij}}{r_{ij}}+\frac{r_{mi}^2+(\mathrm{\α}-2){(r_{mi}\·{\hat{r}}_{ij})}^2}{2r_{ij}^2}\right)\]---\left(15\right)$

$R_n^{\left(\mathrm{\α}\right)}=r_{ij}^{\mathrm{\α}}\[\frac{1}{2}+\mathrm{\α}\left(\frac{r_{nj}\·{\hat{r}}_{ji}}{r_{ij}}+\frac{r_{nj}^2+(\mathrm{\α}-2){(r_{nj}\·{\hat{r}}_{ji})}^2}{2r_{ij}^2}\right)\]---\left(16\right)$

Then formula (11) (12) can be to be write as polymerization, the form shifting, configuring.

5th step, matrix equation solves and the calculating of electromagnetic scattering parameter.

The mode utilizing time recursion solves the current coefficient in each moment.Now scattered field the most spatially is by sources divided into two parts: a part is produced by the source near field group NFP (α) of this site place group；Another part is produced by the source in far field group FFP (α) of this site place group.The recurrence formula of temporal basis functions fast algorithm based on Taylor series expansion becomes:

$Z^0{I}^i=V^i-\underset{{\mathrm{\α}}^{\′\∈NFP\left(\mathrm{\α}\right)}}{\Σ}\underset{j=0}{\overset{i-1}{\Σ}}{Z}_{i-j}^{{\mathrm{\α\α}}^{\′}}{I}_j^{{\mathrm{\α}}^{\′}}-\underset{{\mathrm{\α}}^{\′}\∈\mathrm{FF}P\left(\mathrm{\α}\right)}{\Σ}\underset{n\∈{\mathrm{\α}}^{\′}\left(n\right)}{\Σ}\⟨{\mathrm{\Λ}}_m\left(r\right),{\mathrm{\αE}}_{n,l}(r,t)+(1-\mathrm{\α})n_m\×H_{n,l}(r,t)\⟩---\left(17\right)$

The contribution that near-field region produces is to be calculated by classical MOT algorithm, and main being multiplied with current coefficient by matrix element value is obtained.In far-field region, the namely far field group pair in corresponding group, the interaction between these groups, cross Taylor series expansion and become polymerization-transfer-collocation method quickly to calculate.

Due to the I before moment i Δ t^jIt is all known at moment i Δ t, j=1,2,3 ..., i-1, the most each time step solves the matrix equation of expression of first degree (17), it is possible to obtain the I on each moment i Δ tⁱSo the current value in each moment can be obtained with recursion.Finally can go out, according to the transient current coefficient calculations tried to achieve, the electromagnetic scattering parameter that we need.

In order to verify correctness and the effectiveness of the present invention, analyze below simple guided missile model Electromagnetic Scattering Characteristics.

Example: simple guided missile model, physical dimension is long 8m, radius 0.25m.Driving source is set to: modulation Gaussian pulse, mid frequency 150MHz, bandwidth 300MHz, θ=180 °, direction of incidence wave,(irradiation of bullet direction), polarization mode VV polarizes, 0 °≤θ≤180 ° of viewing angle,Guided missile model uses 0.1 λ (λ=1m) subdivision to obtain 8034 trianglees, and total unknown quantity is 12051.Packet size is 0.3 λ.It is far-field region outside 3 boxes, near field thresholding 0.3 λ * 3.Use Δ t=333.3ps, calculate altogether 600 time steps.Parameter alpha=0.5 of CFIE, uses GMRES alternative manner to calculate, and convergence precision is 1e-9.Fig. 4 gives the guided missile model dual station RCS curve when 30MHz, 150MHz and 270MHz and contrasts with classical MOT algorithm result of calculation, and form 1 has added up the method for this patent proposition and the time of traditional time-domain integration method consumption and internal memory contrast.

Table 1. calculates time and memory consumption contrast

As can be seen from Figure 4 the inventive method calculates dual station RCS data and the result that MOT calculates are the most identical, it was demonstrated that the accuracy of the inventive method.As seen from Table 1, use the inventive method to compare classical MOT algorithm and need preferably to calculate internal memory and calculating time.Further demonstrate the effectiveness of the inventive method.

The present invention to realize process simple, compared to operations such as the complicated time-frequency domain conversion needed, k-space integrals, the inventive method only need to be on the basis of Octree packet and the nearly far field of multilamellar divide, to far field group between distance R carry out Taylor expansion and reconstruct polymerization, the operation shifted, project, program relatively easy being easily achieved.And the inventive method suitability is high, only can need to make less change and just can be applicable to solve medium, in the analysis that has the challenge such as consumption, dispersion.

Claims (2)

1. a temporal basis functions fast algorithm based on Taylor series expansion, it is characterised in that step is as follows:

The first step, sets up time-domain electromagnetic integral equations；Incident electromagnetic field and scattering field is utilized to meet in metal surface Boundary condition set up time-domain electromagnetic integral equations；

Second step, by the sub-scattering object packet obtained discrete on scattering object surface；With a cuboid box by whole target Object surrounds, and this cube is divided into 8 sub-cubes, then each sub-cube is divided into 8 less Cube, the like, until reaching the threshold value pre-set, stop dividing；Mutual between the sub-scattering object of any two Coupling or self coupling be divided into according to their position relationship near field effect to and far field effect right；When they are that near field effect is right Time, calculate near field impedance matrix, when they are for far field effect pair, use Taylor series expansion to become polymerization-transfer-configuration Method calculates the field produced at the on the scene group of basic function of group basic function of source；

3rd step, calculates near field impedance matrix, by metal surface electric current density space basic function and time base function expansion, And in spatial domain, carry out the gold test of gal the Liao Dynasty, time domain carries out Point matching and obtains matrix element value；

4th step, far field effect between use Taylor series expansion reconstruct polymerization, the form calculus source shifted, configure The field produced at group on the scene group of basic function of basic function；

5th step, matrix equation solves and the calculating of electromagnetic scattering parameter；When the mode utilizing time recursion solves each The current coefficient carved, uses solution by iterative method to go out final faradic current coefficient, finally according to the transient current coefficient tried to achieve Calculate the electromagnetic scattering parameter of needs.

Temporal basis functions fast algorithm based on Taylor series expansion the most according to claim 1, its feature exists In: in described step 4, Octree packet and the nearly far field of multilamellar divide on the basis of, to far field effect between enter Row Taylor series expansion reconstruct polymerization, shift, the operation that projects is taken advantage of to accelerate matrix vector；

If the space basic function at source point r' place is Λ_n(r'), the source signal J at r'_n(r', t) is unfolded as follows:

$J_n(r^{\′},t)=\underset{j=1}{\overset{N_t}{\mathrm{\Σ}}}{I}_{n,j}{\mathrm{\Λ}}_n\left(r^{\′}\right)T_j\left(t\right)={\mathrm{\Λ}}_n\left(r^{\′}\right)g_n\left(t\right)---\left(1\right)$

Wherein space basic function Λ_nR () is RWG basic function, time basic function T_jT () is Based on Triangle Basis, N_tIt it is time base letter The number of number；

Source signal J_n(r' t) is broken down into L section continuous print subsignal J_n,l(r', t), the persistent period of each cross-talk signal is T_s=(M_t+ 1) Δ t, M_tFor the length of every cross-talk signal, source signal is written as form:

$J_n(r^{\′},t)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{J}_{n,l}(r^{\′},t)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{\mathrm{\Λ}}_n\left(r^{\′}\right)g_{n,l}\left(t\right)---\left(2\right)$

The test electromagnetic field that at source point r', l cross-talk signal produces at site r is:

$\begin{array}{c}\⟨{\mathrm{\Λ}}_m\left(r\right),n_m\×H_{n,l}(r,t)\⟩\\ =\frac{1}{4\mathrm{\π}}{\∫}_{S_m}{\mathrm{\Λ}}_m\left(r\right)\·n_m\×\left\{{\∫}_{S_n}{\mathrm{dS}}^{\′}\right\{\left[{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right){\mathrm{\Λ}}_n\left(r^{\′}\right)\right]\×\frac{R}{{\mathrm{cR}}^2}+\left[g_{n,l}\left(\mathrm{\τ}\right){\mathrm{\Λ}}_m\left(r^{\′}\right)\right]\×\frac{R}{R^3}\left\}\right\}\mathrm{dS}\\ \≈\frac{1}{4\mathrm{\π}}({\∫}_{S_m}{\mathrm{\Λ}}_m\left(r\right)\mathrm{dS}\·n_m\×{\∫}_{S_n}{\mathrm{\Λ}}_n\left(r^{\′}\right){\mathrm{dS}}^{\′}\×R)[\frac{1}{c{R}^2}{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right)+\frac{1}{R^3}{g}_{n,l}\left(\mathrm{\τ}\right)]\\ =\frac{1}{4\mathrm{\π}}(m_m\·n_m\×m_n\×R)[\frac{1}{c{R}^2}{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right)+\frac{1}{R^3}{g}_{n,l}\left(\mathrm{\τ}\right)]\end{array}---\left(3\right)$

$\begin{array}{c}\⟨{\mathrm{\Λ}}_m\left(r\right),E_{n,l}(r,t)\⟩\\ ={\∫}_{S_m}{\∫}_{S_n}{\mathrm{\Λ}}_m\left(r\right)\·{\mathrm{\Λ}}_n\left(r^{\′}\right)\frac{\mathrm{\μ}{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right)}{4\mathrm{\πR}}{\mathrm{ds}}^{\′}\mathrm{ds}+{\∫}_{S_m}{\∫}_{S_n}\▿\·{\mathrm{\Λ}}_m\left(r\right)\▿\·{\mathrm{\Λ}}_n\left(r^{\′}\right)\frac{{\∂}_{\mathrm{\τ}}^{-1}{g}_{n,l}\left(\mathrm{\τ}\right)}{4\mathrm{\π\ϵR}}{\mathrm{ds}}^{\′}\mathrm{ds}\\ \≈{\∫}_{S_m}{\mathrm{\Λ}}_m\left(r\right)\mathrm{ds}\·{\∫}_{S_n}{\mathrm{\Λ}}_n\left(r^{\′}\right){\mathrm{ds}}^{\′}\frac{\mathrm{\μ}{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right)}{4\mathrm{\πR}}+{\∫}_{S_m}\▿\·{\mathrm{\Λ}}_m\left(r\right)\mathrm{ds}{\∫}_{S_n}\▿\·{\mathrm{\Λ}}_n\left(r^{\′}\right){\mathrm{ds}}^{\′}\frac{{\∂}_{\mathrm{\τ}}^{-1}{g}_{n,l}\left(\mathrm{\τ}\right)}{4\mathrm{\π\ϵR}}\\ =\frac{\mathrm{\μ}}{4\mathrm{\π}}{m}_m\·m_n\frac{{\∂}_{\mathrm{\τ}}{g}_{n,l}\left(\mathrm{\τ}\right)}{R}\±\frac{l_m{l}_n}{4\mathrm{\π\ϵ}}\frac{{\∂}_{\mathrm{\τ}}^{-1}{g}_{n,l}\left(\mathrm{\τ}\right)}{R}\end{array}---\left(4\right)$

Wherein Λ_mR () is the test basic function at site r, n_mFor the outer normal vector of the unit at site r, E_n,l(r,t),H_n,l(r, t) is electromagnetism and the magnetic field that at source point r', l cross-talk signal produces at site r, μ and ε is respectively The pcrmeability of free space and dielectric constant；R=| r-r'|, c are the lighies velocity in free space, and τ=t-R/c is to prolong Time；l_m,l_nIt is respectively the length of side on m and n article of limit,Exist for basic function Integration in its supporting domain；

Source point r_nWith site r_mLaying respectively in two groups, two groups are called source group and field group, and group switching centre is respectively r_i And r_j, the vector representation between field source basic function is:

R=r_mi+r_ij-r_nj=R_m-R_n (5)

Here, r_ij=r_i-r_j, r_mi=r_m-r_i, r_nj=r_n-r_j R_m=r_mi+r_ij/ 2, R_n=r_nj-r_ij/2

Taylor series expansion is utilized to obtain following expression:

$\begin{array}{c}{R}^{\mathrm{\α}}={(R\·R)}^{\frac{\mathrm{\α}}{2}}={[(r_{\mathrm{mi}}+r_{\mathrm{ij}}-r_{\mathrm{nj}})\·(r_{\mathrm{mi}}+r_{\mathrm{ij}}-r_{\mathrm{ni}})]}^{\frac{\mathrm{\α}}{2}}\\ =r_{\mathrm{ij}}^{\mathrm{\α}}{[1+(\frac{{2r}_{\mathrm{mi}}\·{\hat{r}}_{\mathrm{ij}}}{r_{\mathrm{ij}}}+\frac{r_{\mathrm{mi}}^2}{r_{\mathrm{ij}}^2})+(\frac{{2r}_{\mathrm{nj}}\·{\hat{r}}_{\mathrm{ji}}}{r_{\mathrm{ij}}}+\frac{r_{\mathrm{nj}}^2}{r_{\mathrm{ij}}^2})\frac{2r_{\mathrm{mi}}\·r_{\mathrm{nj}}}{r_{\mathrm{ij}}^2}]}^{\frac{\mathrm{\α}}{2}}\\ \≈R_m^{\left(\mathrm{\α}\right)}+R_n^{\left(\mathrm{\α}\right)}\end{array}---\left(6\right)$

$R_m^{\left(\mathrm{\α}\right)}=r_{\mathrm{ij}}^{\mathrm{\α}}[\frac{1}{2}+\mathrm{\α}(\frac{r_{\mathrm{mi}}\·{\hat{r}}_{\mathrm{ij}}}{r_{\mathrm{ij}}}+\frac{r_{\mathrm{mi}}^2+(\mathrm{\α}-2){(r_{\mathrm{mi}}\·{\hat{r}}_{\mathrm{ij}})}^2}{2r_{\mathrm{ij}}^2})]---\left(7\right)$

$R_n^{\left(\mathrm{\α}\right)}=r_{\mathrm{ij}}^{\mathrm{\α}}[\frac{1}{2}+\mathrm{\α}(\frac{r_{\mathrm{nj}}\·{\hat{r}}_{\mathrm{ji}}}{r_{\mathrm{ij}}}+\frac{r_{\mathrm{nj}}^2+(\mathrm{\α}-2){(r_{\mathrm{nj}}\·{\hat{r}}_{\mathrm{ji}})}^2}{2r_{\mathrm{ij}}^2})]---\left(8\right)$

Then (5-8) is substituted into formula (3) (4) by write as polymerization, shift, the form that configures is taken advantage of to accelerate matrix vector.